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# Download PowerPoint Presentation - An Introduction to Differential Equations

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An Introduction to Stress and Strain Lecture 9/14/2009 GE694 Earth Systems Seminar Some definitions: • Body force == A force that acts throughout an object. Example: gravity is a body force because gravity affects every every single atom in an object in the same way. • Pressure and stress == force per unit area on the surface of an object. Example: air pressure acts on the surface of one’s body. Pressure and stress can be defined on imaginary surfaces as well as on real surfaces. Some mathematics: ^ • A force F acts in some direction n with magnitude |F|. It can be described by a vector F=(Fx, Fy, Fz). y Fa Fb Fa is a stronger force than Fb, and it acts in a different direction. x F Fy Fx • A stress nm is a force that acts in some direction n^ with magnitude | nm | on a surface where a normal vector to the ^ surface points in direction m. The complete description of stress at a point is described by a matrix of 9 values. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. Stress Intensity of Force Stress = Force /Area • Not all nine components of a stress tensor (matrix) are independent. Since xy=yx, yz=zy, and xz=zx, there are only 6 independent values. • A stress tensor can be divided into two tensors. One tensor is the isotropic tensor, which is also called the hydrostatic stress tensor. The other tensor is the deviatoric stress tensor. Isotropic stress (pressure) QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. ( p p ) Deviatoric stress QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. ( 'xx 'xy 'xz ’yx ’yy ’yz ’zx ’zy ’zz ) • Normal stresses -- xx, yy and zz. The force is oriented in the same direction as the normal to the surface. Normal stresses can compress or stretch an object, changing its volume. Angles cannot be changed by normal stresses. (Figure A) • Shear stresses -- xy, xz, yz. The force is oriented parallel to the surface. Shear stresses can deform objects by changing angles in the object (like shearing a deck of cards). Volume cannot be changed by shear stresses. (Figure B) • It is very difficult to determine the absolute value of stress in the Earth. However, there are many ways that the relative values of stress in different directions can be estimated. It is also possible to estimate the changes of stress with time. • Stresses can cause objects to deform through elastic behavior (spring-like behavior) or through fluid flow. High values of shear stresses can cause brittle objects to fracture. • Strain nm is a deformation of an object due to an applied stress. The complete description of strain at a point is described by a matrix of 9 values. Strain refers to a change in length per unit length (L/L ) or to a change in angles due to an applied stress. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. Shear strains (describe changes in angles) QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. Normal strains (describe length and volume changes) Normal strains (describe length and volume changes) Dilatation=change in volume per unit volume Shear strains (describe changes in angles) • Strains can be found by taking spatial derivatives of displacements. If a displacement changes with position, then there is a strain. Let wx, wy and wz be the displacements in the x, y and z directions, respectively. Then the following expressions show how elements of the strain tensor are computed from spatial derivatives of displacements. Shear strains (describe changes in angles) QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. Normal strains (describe length and volume changes) QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. Dilatation=change in volume per unit volume • For infinitesimal strains, not all nine components of a strain tensor (matrix) are independent. Since xy=yx, yz=zy, and xz=zx, there are only 6 independent values. • A strain tensor can be divided into two tensors. One tensor is the isotropic tensor, which describes volume changes. The other tensor is the deviatoric strain tensor, which describes angle changes (shear). Isotropic strain (volume change)QuickTime™ and a TIFF (LZW) decompressor are needed to see this and picture. QuickTime™ a TIFF (LZW) decompressor are needed to see this picture. ( e e e ) (e is the dilatation divided by 3) Deviatoric strain (shear) QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. ( 'xx 'xy 'xz ’yx ’yy ’yz ’zx ’zy ’zz ) • Since strain is determined by spatial changes in displacements, it must always be referenced to the condition when the displacements were 0. For many deformation situations, one determines the strain rate, or the change of strain with time. Strain rate example QuickTi me™ and a T IFF (LZW ) decompressor are needed to see t his pict ure. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. • The relationship between stress and strain is described by a “constitutive law”. There are many constitutive laws, each one corresponding to a different kind of material behavior. Some kinds of material behaviors that have their own constitutive laws: – Elastic material – Newtonian viscous fluid – Non-Newtonian viscous fluid – Elastoplastic material An Introduction to Fluid Mechanics Lecture 9/14/2009 GE694 Earth Systems Seminar Fluid Mechanics • Fluids flow due to changes in pressure with distance. Fluids always flow from areas of high pressure to areas of low pressure. The amount of fluid flow is controlled by the pressure gradient over distance l, defined as (p0-p1)/l, where p0 and p1 are the pressure values separated by distance l. The flow of a fluid depends on the pressure gradient, the properties of the fluid, drag conditions that affect the flow, and the existence or absence of turbulence. A Simple Fluid Flow Problem: Channel Flow • The flow of a fluid in a channel illustrates some basic properties of fluid flows. In this problem, there is a flow through a channel where one boundary of the flow is stationary and the other moves with the flow. This could represent the flow of water in a river. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. Some basic fluid dynamics concepts: • “Newtonian” fluid flow law: QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. Note: Equation 6-1 is a constitutive law for a Newtonian viscous fluid. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. u is the velocity of the fluid (displacement with time), so du/dy is a shear strain rate. is a shear stress acting on the fluid. is called the dynamic viscosity. For a viscous fluid, shear stress causes the fluid to flow. is the density of the fluid • The force balance equation (i.e., Newton’s second law) is also called the equation of motion. For channel fluid flow being driven by a pressure gradient (p0-p1)/l, the equation of motion is: QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. Hydraulic head has units of distance (g is the acceleration due to gravity). QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. • The solution for the velocity in the channel flow problem is: Equation (6-10) is the equation of motion for this problem. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. Here u0 is the velocity at the freely moving surface at the top of the fluid. • A drawing of the solution to this problem is: QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. Note how the flow must go to 0 at the base, since a viscous fluid in contact with a solid boundary must have the same velocity as the boundary. The solid boundary base acts to resist the flow, and the viscous flow imposes a stress on the top of the solid base. • Simple variations on this problem: The top plot corresponds to a situation where there is no pressure gradient but the top of the flow is being driven at velocity u0. The bottom plot corresponds to a situation where there is a pressure gradient but the top of the flow as well as the bottom of the flow have no velocity. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. Pipe Flow Problem: • This problem is similar to the channel flow problem. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. • Laminar versus turbulent flow: QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. Note: The previous solutions that we have looked at all assume laminar flows. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. • The general properties of fluid flows can be estimated using dimensionless variables: Theory of Fluid Flow • The quantity that is normally measured in fluid flow problems is the fluid velocity. Let velocity be a vector (u, v, w) where u is the velocity in the x direction, v is the velocity in the y direction, and w is the velocity in the z direction. For a steady flow, the amount of fluid flowing into any volume is the same as the amount of fluid flowing out of any volume (conservation of mass). This is described mathematically by the “continuity equation”. Continuity Equation: QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. Note: The above is a 2-dimensional form of the continuity equation for an imcompressible fluid. In 3-D, the continuity equation is (u/x)+(v/y)+(w/z)=0 Equation of Continuity Compressible or Incompressible Fluid Flow Most liquids are nearly incompressible; that is, the density of a liquid remains almost constant as the pressure changes. To a good approximation, then, liquids flow in an incompressible manner. In contrast, gases are highly compressible. However, there are situations in which the density of a flowing gas remains constant enough that the flow can be considered incompressible. Setting up and solving general fluid flow problems: A simple example • The following is a simple 2-D example of how a fluid flow problem is set up to find the differential equations that need to be solved. In this simple example, there is a steady-state flow (it does not change with time) in the x and y directions. Gravity acts in the y direction. • The starting equations: The constitutive law for a Newtonian fluid (in 2-D), including the pressure that drives the flow QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. (xy=xy) QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. The continuity equation for an incompressible fluid in 2-D is QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. Combine the constitutive law and the continuity law equations to get the force-balance equations (i.e., Newton’s second law) for this simple flow. The result is a simple form of the “Navier-Stokes Equation”: QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. • Solving the equations: Equation pairs 6-64 and 6-65 or 6-66 and 6-67 are the ones we want to solve for the flow (i.e., the velocities u and v). A common way to solve these is to solve for a stream function where QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. The solutions for the stream function, and hence for u and v, can only be completely determined once the boundary and initial conditions are specified. After the solution for the stream function is found, the velocities u and v are found by taking the above derivatives of the stream function. Streamline Flow When the flow is steady, streamlines are often used to represent the trajectories of the fluid particles. Streamlines are defined by the stream function . A streamline is a line drawn in the fluid such that a tangent to the streamline at any point is parallel to the fluid velocity at that point. Steady flow is often called streamline flow. (a) In the steady flow of a liquid, a colored dye reveals the streamlines. (b) A smoke streamer reveals a streamline pattern for the air flowing around this pursuit cyclist, as he tests his bike for wind resistance in a wind tunnel. Stokes Flow • The problem of a solid spherical object being driven up or down in a fluid due to buoyancy forces (gravity and density differences) is known as Stokes flow. The solution can be applied to many different kinds of problems in the geosciences. • This problem is best solved in spherical coordinates, i.e. (, . QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. • Here is the geometry of the problem QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. The equations of motion are QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. The continuity equation for an incompressible fluid is • The starting equations: QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. The continuity equation and the equations of motion can be manipulated into the forms QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. We look for solutions of the form After solving the equations and applying the boundary conditions, the final solutions for the velocity components are QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. There is drag on the sphere, which is a combination of the pressure drag Dp and the viscous drag Dv. These two drags are QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a TIFF (LZW ) decompressor are needed to see this QuickTime™ andpicture. a TIFF (LZW) decompressor are needed to see this picture.